Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) = C24×#120 (order 512) · Inn(G) = #120 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | C2×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10)(3,11,13,16)(4,9,5,8)(7,14,15,12)(17,18,22,20)(19,25,28,32)(21,29,31,24)(23,30,26,27);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,11)(10,15)(17,19,22,28)(18,23,20,26)(21,30,31,27)(24,32,29,25);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,20,22,18)(19,26,28,23)(21,24,31,29)(25,30,32,27)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=g,d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,d^b=df,e^b=eg,d^c=dg,e^c=efg20 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1664, #18987, #18990, … |
| maximal normal subgroup of | C2×#1664, #18987, #18990, … |
| maximal quotient of | #7764, #7765, #7766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 396);